The task of joint channel and data estimation without a training sequence is of high importance for Maximum Likelihood Sequence Estimation (MLSE) processing, as widely described in prior art. MLSE processing is suitable for fiber optical communication systems. Especially, due to the presence of Polarization Mode Dispersion (PMD—modal dispersion where two different polarizations of light in a waveguide, propagate at different speeds, causing random spreading of the optical pulses), the optical channel is considered to be non-stationary and adaptive equalization is required. When the histograms, which serve as channel estimators, are updated faster than the channel variation rate, successful variations tracking can be achieved.
Maximum Likelihood Sequence Estimation (MLSE) is considered to be a non-linear equalization technique. To explain the main idea that is behind the MLSE processing, trellis diagrams are often used. The example of 4-state trellis diagram is presented in FIG. 1a. In this example, two bits of the channel memory is assumed, i.e., current sample is affected by two previous bit and the current bit. The two previous bits define the channel state, and a conjunction of a state with a current bit defines a branch. The arrows in the trellis represent the transition from one state to another, while a transition that corresponds to a “0” bit is drawn with solid line and a transition corresponding to a “1” is represented by a dashed line. To each branch in the trellis the number called “branch metric” is assigned, which depends on channel and noise statistics. Branch metric describes in some sense the probability of a corresponding transition. Furthermore, one can define a “path metric” which is the sum of corresponding branch metrics for a certain path, when a bit sequence with a smallest path metric is the most probable to be transmitted.
An MLSE processor chooses the path with the smallest metric, and produces the most likely sequence by tracing the trellis back. For a sequence of length N, there are 2N possible paths in the trellis. Therefore, an exhaustive comparison of the received sequence with all valid paths is a cumbersome task, becoming non feasible for channels with a long memory. However, since not all paths have the similar probabilities (or metrics) when proceeding through the trellis, there is an efficient known algorithm, called Viterbi algorithm (is a dynamic programming algorithm for finding the most likely sequence of hidden states), that limits the comparison to 2K “surviving paths”, where K is the channel memory length, independent of N, making the maximum likelihood principles to be practically feasible.
The Viterbi Algorithm
From two paths entering the trellis node, the path with the smallest metric is the most probable. Such a path is called the “surviving path”, and only surviving paths with their running metrics need to be stored.
Maximum likelihood sequence detection is the most effective technique for mitigating optical channel impairments, such as Chromatic Dispersion (CD—the dependency of the phase velocity of an optical signal on its wavelength) and Polarization Mode Dispersion (PMD). In order to successfully apply this technique, it is mandatory to estimate some key channel parameters, needed by the Viterbi processor.
Conventional channel estimation methods can be classified as parametric and non-parametric. Parametric methods assume that the functional form of the Probability Density Function (PDF—a function that describes the relative likelihood for this random variable to take on a given value) is known, and only its parameters should be estimated. However, non-parametric methods do not assume any knowledge about the PDF functional form or its parameters. There are two most common methods, used in practice for channel estimation: Method of Moments (MoM—a way of proving convergence in distribution by proving convergence of a sequence of moment sequences) and Histogram Method (of estimation).
Method of Moments is considered to be parametric, and therefore, it assumes that the functional form of the PDF is known and only its moments need to be estimated. When the dominant noise mechanism in the optical system is thermal, like in optically unamplified links, the conditional PDF of the received sample xn, given that μk is transmitted, is assumed to be Gaussian with σn2 being the variance of the noise:
                                          f            channel            Gaussian                    ⁡                      (                                          x                n                            ❘                              μ                k                                      )                          =                              1                                          2                ⁢                                  πσ                  n                  2                                                              ⁢          exp          ⁢                      {                          -                                                                    (                                                                  x                        n                                            -                                              μ                        k                                                              )                                    2                                                  2                  ⁢                                      σ                    n                    2                                                                        }                                              [                  Eq          .                                          ⁢          a                ]            
In this case, only first and second moments need to be estimated. Another case of interest is Amplified Spontaneous Emission (ASE) limited channel. The noise in such a channel becomes signal dependent and the functional form of the conditional PDF of the received sample xn, given that μk is transmitted, can be approximated by a non-central Chi-square distribution with v degrees of freedom:
                                          f            channel            ASE                    ⁡                      (                                          x                n                            ❘                              μ                k                                      )                          =                              1                          N              0                                ⁢                                    (                                                x                  n                                                  μ                  k                                            )                                      (                                                v                  -                  1                                2                            )                                ⁢          exp          ⁢                      {                          -                                                                    x                    n                                    +                                      μ                    k                                                                    N                  0                                                      }                    ⁢                      I                          v              -              1                                ⁢                      {                          2              ⁢                                                                                          x                      n                                        ⁢                                          μ                      k                                                                                        N                  0                                                      }                                              [                  Eq          .                                          ⁢          b                ]            
Where I{•} is the modified Bessel function of the first kind and N0 is power spectral density of the ASE noise given by:
      N    0    =            n      sp        ⁢          hc              λ        0              ⁢          (              G        -        1            )      where nsp is the spontaneous emission factor (or population inversion factor), G represents the EDFA gain,
  hc      λ    0  is the photon energy at the wavelength λ0, h being the Plank constant and c being the speed of light. It is clear that in Eq. b, N0 and v need to be estimated.
Histogram Method
The histogram method does not assume anything about the PDF of the received samples. According to this method MNisi histograms are collected, where M represents the vocabulary size of the transmitted symbols and Nisi is the number of the most resent previous symbols that affect the current symbol, i.e., the channel memory length assumed by the algorithm. The received signal is assumed to be quantized to NADC bits; Therefore, each histogram consists of 2NADC bins (discrete intervals), where NADC is a design parameter. Each histogram can be uniquely associated with a branch in the trellis diagram of the receiver. Assuming that the number of signal samples collected is large, the histogram (normalized so that the sum of all its bins is unity) is an estimate of fchannel(xn|μk). The histogram is updated iteratively, based on the observed samples and the decision bits at the output of the MLSE decoder.
The branch metrics are obtained by taking the natural logarithm of the estimated/assumed PDF. For a transmitted sequence of length N the MLSE decoder chooses between MN possible sequences that minimize the (path) metric:
                              m          r                =                              ∑                          n              =              1                        N                    ⁢                                    -              ln                        ⁢                          {                                                f                  channel                                ⁡                                  (                                                            x                      n                                        ❘                                          μ                      k                                                        )                                            }                                                          [                  Eq          .                                          ⁢          c                ]            
The estimated bit sequence is determined by tracing the trellis back, based on the minimal path metric of Eq. c.
In optical fiber systems, the purpose of the MLSE is to overcome ISI stemming from CD and from PMD. While CD is a deterministic phenomena for a given link, PMD is stochastic in nature, and therefore, an adaptive equalizer that performs PMD tracking is required. Moreover, the adaptation properties of the MLSE can be also exploited for CD compensation when the amount of CD is not accurately known. Basically, expensive tunable optical dispersion compensation may be replaced by the adaptive MLSE. This type of operation, without knowing any initial information about the channel parameters and distortion is called “blind equalization”.
The constant growth in the demand for high bandwidth data transmission leads to higher challenges that should be resolved in the physical layer, and particularly by optical transmission technology.
The current high end transmission data rates are in the range of hundreds of Gbits/sec. One emerging technology that can support such bitrates for long distances (hundreds of kilometers and above) is coherent transmission and detection. On the other hand, direct detection technology offers the use of lower cost optoelectronic components, consumes less power and enables overall lower latency solution. These advantages may be critical for short reach applications such as sub-hundred kilometers networks of metro-edge and data centers interconnections.
The simpler alternative, (non-coherent) direct detection optical technology is of lower cost, but is limited to lower bit rates and/or shorter distances. For example, increasing the bitrate from 10 Gbit/sec to 25 Gbit/sec, results in distance reduction from ˜80 km to ˜15 km, for the same Bit Error Rate (BER) performance. The main reason for this limitation is the inter-symbol interference (ISI—a form of distortion of a signal in which one symbol interferes with subsequent symbols) caused by chromatic dispersion (CD) and Polarization Mode Dispersion (PMD). Two techniques are commonly used in order to overcome this ISI. The first technique is based on advanced modulation formats, together with partial response signaling, while the other approach is based on Digital Signal Processing (DSP), applying Electronic Dispersion Compensation (EDC—a method for mitigating the effects of chromatic dispersion in fiber-optic communication links with electronic components in the receiver). The EDC implementations with Maximum Likelihood Sequence Estimation (MLSE) at the receiver (Rx) side, is theoretically the optimal tool to combat ISI, this was very popular for 10 Gbis/sec. The combination of the two approaches is also possible, and was also theoretically investigated for 4×25 Gbits transmission with the use of reduced bandwidth components.
The prior art methods described above for performing equalization of an optical channel are not suitable for blind equalization, since they either require training sequence, or have a low convergence rate, or involve high implementation complexity. Moreover, they require using real data signals to converge, which results in a relatively high initial Bit-Error-Rate (BER—the percentage of bits that have errors relative to the total number of bits received in a transmission) and data loss.
Also, blind channel estimation for the MLSE receiver for direct detection systems allows upgrading the current 10 G systems to 100 G (4×25 G) systems, with extended reach of up to 40 km uncompensated links. The task of joint channel and data blind estimation without a training sequence in hand for optical communication with direct detection is of high importance for MLSE processing. Most of the prior art methods deal with a steady state operation, i.e. the tracking mode. However, the blind estimation of the optical channel suitable for the acquisition/initialization stage is less covered. Although various MLSE acquisition methods exist, most of them either require a training sequence, or have a low convergence rate, or involve high implementation complexity.
It is therefore an object of the present invention to provide a method for performing blind equalization of an optical channel without requiring a training sequence.
It is another object of the present invention to provide a method for performing blind equalization of an optical channel with a high convergence rate.
It is still another object of the present invention to provide a method for performing blind equalization of an optical channel which is easy to implement.
It is a further object of the present invention to provide a method for performing blind equalization of an optical channel which does not require using real data signals to converge.
Other objects and advantages of the invention will become apparent as the description proceeds.